Outside of cost, there are several factors that distinguish white gold and platinum. White gold has a beautiful bright white patina, achieved by rhodium plating, which periodically must be re-plated in order to maintain the bright white luster. Platinum does not have any coating and will build a natural patina over time.
In terms of durability, platinum is a dense metal and more durable to wear and tear than white gold, which is more prone to damage because of the alloys used in creating white gold.
Finally, platinum is a heavier metal, with white gold being lighter and better suited to people who may not desire the weight of platinum.